The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator

Laurent  Denis; Anis  Matoussi; Jing  Zhang

doi:10.3934/dcds.2015.35.5185

Article Contents

2015, Volume 35, Issue 11: 5185-5202. Doi: 10.3934/dcds.2015.35.5185

This issue Previous Article Global existence for the stochastic Degasperis-Procesi equation Next Article Invariant foliations for stochastic partial differential equations with dynamic boundary conditions

The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator

1.
Institut du Risque et de l'Assurance, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9
2.
School of Mathematical Sciences, University of Fudan, Handan Road 220, 200433, Shanghai

Received: October 31, 2012

Revised: April 30, 2014

Published: October 2015

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Abstract

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,\nu)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of Itô's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.

Keywords:

Parabolic potential,
regular measure,
stochastic partial differential equations,
non-homogeneous second order operator,
obstacle problem,
penalization method,
Itô's formula,
comparison theorem,
space-time white noise.

Mathematics Subject Classification: 60H15, 35R60, 31B150.

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